Question

Give 2 different proofs that F is conservative. F = (ycos(x) + y, sin(x) + x) Provide at least one theorem or equation or formula number from the sheet I provided you in your solution. Make it clear where the theorem/equation is being used.

Answer 1

PAGE 1

^ 1 For a giver, F = <ycos(x) + y, sin (x) +X ? = (y cosx + y)i + (sinx+x) differentiable , 3-dimensional vector field, 7:18°— R’, we can say F is conseudue, then curl F = 0 JF OF, = 0 where dy EF E continuously Эх + Hence, for ² = (y losx (ylosx+y) + (sisx+x)) Fi = y losx + y ; Fa = Sinata : 8F (sinx+x)= δα Cost 1 ax OF a lylos x+y) = cosx + 1 ay ay OF, Cosxt 1-losx-1=0 200 ду - ƏF, Hence, F' is a conservature field

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b) For a vector field to be conservative, fF.de = f(b) fta) where cis any audulechy home with a bas endpoints, Here, f(x,y) is a potential function swkihat (b) = (06, 9), (Q) = (xa, ya) TE=alf Funding the potentid function fa,y); gues, ît sinxtx F(x,y) = y cosxey (P ΣΕ ay of, df doc dal - Jet tosx+1 2oC from condition, If = ² => (of, [F, , Eq) ди. y cosx +y = f(x,y) = fylosx+y treating foxy) = y sinx ty xt sky) yasa anst. Cecy) Sunce y was treated as a const) Enter colonial] Similarly, - SmX+ x => Aerod (ysis x +yx+8(y)) xe te he he 2 sinxex

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= Six XtX BUD X + OC + 2 (96)) dy 27 d (819) o ay => sly) - K (where K is some constant) f(x,y) functions sit. y sin x + y x + K is a potential If y q (0,1) Die co-D Tessilinig LP. doi let C: x= cost &g= sint 7 st = 1 2 3 ?do= I cycosity) ? Sun X+3) j). (dzî+dýl (senx+20dy) - 0 C C {[(y cosx+y) dx + t Surie, x= cost da -- sint=7 dx=-sintalt at y=sint 7 dy cost » dy= costat It

PAGE 4SCREENSHOT:

& substlikime te paramilie values of Xoy & deady in D (FJ. ( Goint cocot)esit) dhe siste f (suis lost) + (tostot -1/2 + -1/2 1/2 SE sis?t costlost) - sint + cost sis (cost) + 605776 1/2 | [cest siollost) -sin?t cos llest) + Cos@+)] dt -1/2 By using Mathematus scaccenshot attached? as, Q = (0-D & b= (0,0) f(0,1) - f(0, -1) = 0 - 0 = 0 fF.de = f(b) f(a) Henu, F is conservative ...

In[3] = ClearAll Out[3]= Clearali In[2] = N( Integrate [cos [t] Sin[Cos[t]] - (Sin[t])? Cos [Cos [t]] + Cos [2t], {t, t, {t, -}] Out[22]= 0. Number Form [0., 16] (*showing all digits*) Out[23]//NumberForm= 0.

SUMMARY:

The proof and Mathematica screenshot has been attached. There is no certain answer that needs to be summarised. The vector field is conservative, is shown in two ways, one by showing it's path independence and the other one by showing its curl is zero, hence it is a gradient of some scalar function (thus, the definition of conservative vector field).